Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge

1. Problem Statement

The paper addresses the well-posedness and higher-order regularity of the inhomogeneous heat equation in an unbounded two-dimensional wedge domain $\Omega$ with Robin boundary conditions.

The problem is formulated as:
$$\begin{cases} \partial_t U - \Delta U = F & \text{in } \mathbb{R}^+ \times \Omega, \\ \gamma U + \partial_\nu U = G & \text{on } \mathbb{R}^+ \times \partial'\Omega, \\ U|_{t=0} = 0 & \text{on } \Omega, \end{cases}$$
where:

• $\Omega = \{ r(\cos \phi, \sin \phi) : r > 0, \phi \in (0, \theta) \}$ is a wedge with opening angle $\theta \in (0, 2\pi)$.
• $\partial'\Omega$ is the boundary excluding the tip (origin).
• $\gamma \in (0, \infty)$ is the Robin parameter.
• The domain is unbounded, and the boundary condition is non-scaling invariant (due to the constant $\gamma$ term).

Key Challenge: Standard elliptic regularity theory fails in non-smooth domains (like wedges) due to singularities at the tip. Furthermore, the lack of scaling invariance in the Robin boundary condition prevents the direct application of the Mellin transform, a standard tool for scaling-invariant wedge problems. The authors must handle "resonances" (specific eigenvalues of the Neumann Laplacian) that occur at specific opening angles.

2. Methodology

The authors develop a novel mathematical framework combining weighted Sobolev spaces, Mellin and Laplace transforms, and an iterative regularity approach.

A. Functional Setting: Weighted Spaces

To handle singularities at the origin and the unbounded nature of the domain, the authors work in monomially weighted Sobolev spaces $H^k_{\beta}(\Omega)$ and parabolic spaces $H^s_{\beta, 0}$.

• Weights: The norms involve a power weight $r^{-\alpha}$ in the radial variable and an exponential weight $e^{-\beta t}$ in time.
• Intersection Spaces: The solution space is an intersection of weighted and unweighted spaces. This allows the authors to first construct a variational solution in an unweighted space (where standard coercivity holds) and then upgrade to weighted spaces to control behavior near the tip and at infinity.
• Resonance Avoidance: The analysis requires excluding specific opening angles where resonances occur. This is formalized by the condition $\frac{\pi}{\theta} \notin \mathbb{Q}$ and distance conditions involving the scaling exponents $s^\Omega_\sigma = \sigma - 1$.

B. The Resolvent Approach

The parabolic problem is treated via its resolvent equation (Laplace transform in time):
$$\mu u - \Delta u = f \quad \text{in } \Omega, \quad \gamma u + \partial_\nu u = g \quad \text{on } \partial\Omega.$$
The authors establish maximal regularity for this elliptic problem for $\mu$ in a sector of the complex plane.

C. The "Test Function" Technique

A central innovation is the use of a test function problem to overcome the lack of scaling invariance.

1. Surjectivity: Instead of testing the resolvent equation directly with arbitrary smooth functions (which might not yield sufficient regularity), they solve an auxiliary problem $Av = w$ (where $A$ is a specific differential operator involving derivatives) to generate a rich class of test functions.
2. Lax-Milgram: They define a sesquilinear form using these test functions. By proving the surjectivity of the operator $A$ onto smooth compactly supported functions, they ensure that the variational solution obtained via the Lax-Milgram theorem is actually a classical solution (satisfying the equation pointwise almost everywhere).
3. Induction: This classical solution serves as the base case for an induction argument to prove higher regularity.

D. Handling Resonances and Singularities

The authors explicitly account for the kernel of the Neumann Laplacian, which contains singular functions of the form $r^{\pi k/\theta} \cos(\frac{\pi k \phi}{\theta})$ and potentially $\ln r$.

• They decompose the solution $u$ into a regular part (in a weighted space) and a singular polynomial part (a finite sum of monomials).
• The "polynomial spaces" $P^\Omega_{k, \beta}$ are introduced to capture these singular behaviors, ensuring the regular part has the necessary smoothness.

3. Key Contributions

1. Framework for Non-Scaling Invariant Problems: The paper provides a systematic method to treat parabolic problems in wedges where the boundary condition breaks scaling invariance, a scenario where standard Mellin transform techniques fail.
2. Arbitrarily High Regularity: Unlike previous works restricted to low regularity or specific angles, this framework establishes arbitrarily high regularity ($\ell \in \mathbb{N}$) for the solution, provided the data is sufficiently smooth and the opening angle avoids resonant values.
3. Uniform Estimates: The estimates derived are uniform with respect to the Robin parameter $\gamma \in (0, \infty)$. The dependence on $\gamma$ is handled via a scaling argument ($S_\gamma(t, x) = (t/\gamma^2, x/\gamma)$), reducing the general case to $\gamma=1$.
4. Resolution of Resonance Issues: The paper rigorously handles the "resonance" phenomenon (where the scaling exponent matches an eigenvalue of the angular operator) by defining precise distance conditions (Eq. 1.2 and 1.3) that exclude these critical cases.

4. Main Results

Theorem 1.1: Well-Posedness (Base Regularity)

• Assumptions: The opening angle $\theta$ satisfies $\frac{\pi}{\theta} \notin \mathbb{Q}$. The weight exponent $\alpha \in (-1, 0)$ satisfies specific distance conditions from resonant values (Eq. 1.2).
• Result: For data $F$ and $G$ in appropriate weighted fractional Sobolev spaces, there exists a unique solution $U$ in the intersection space $E$.
• Estimate: The solution satisfies a bound $\|U\|_E \lesssim \|F\|_F + \|G\|_G$, with constants independent of $\gamma$.

Theorem 1.2: Higher Regularity

• Assumptions: The data $(F, G)$ possesses higher regularity order $\ell$. The parameters $(\alpha, \ell)$ satisfy the non-resonance condition (Eq. 1.3).
• Result: The solution $U$ gains $\ell + 2$ orders of regularity relative to the data. Specifically, if the data is in spaces corresponding to order $\ell$, the solution lies in spaces corresponding to order $\ell+2$.
• Significance: This confirms that the singularities at the wedge tip are fully characterized by the polynomial expansion, and the remainder is as smooth as the data allows.

5. Significance and Impact

• Theoretical Advancement: This work bridges a gap in the theory of parabolic equations on non-smooth domains. While elliptic problems in wedges are well-studied (often via Mellin transforms), the parabolic case with non-scaling invariant boundary conditions (Robin) had lacked a comprehensive high-regularity theory.
• Methodological Innovation: The "test function problem" approach combined with intersection spaces offers a robust template for analyzing other non-scaling invariant problems in singular domains.
• Applications: The results are crucial for modeling physical phenomena involving heat transfer, diffusion, or fluid dynamics in geometrically singular regions (e.g., cracks, corners, or contact lines) where Robin boundary conditions (representing heat transfer or slip) are physically relevant.
• Generality: The authors note that their techniques can be extended to higher-dimensional wedges ($\Omega \times \mathbb{R}^d$) via partial Fourier transforms, suggesting broad applicability beyond the 2D model presented.

In summary, the paper establishes a rigorous, high-regularity theory for the heat equation in a wedge with Robin boundary conditions, overcoming the dual challenges of geometric singularities and non-scaling invariance through a sophisticated blend of weighted analysis and iterative regularity arguments.